The Fourier approach, that is restricting the data to an interval
\(L[-\pi ,\pi ]\)  (\(L=const\) , \(L\in \mathbb {R}^{+}\) ) and continuing them periodically with
period \(2\pi L\)  on the whole real line works very well for periodic
functions and exhibits spectral convergence, namely an
exponential decrease of the numerical error with the number of
Fourier modes. Schwartz class functions can be treated as periodic as
one works with a finite precision and since \(L\)  can be chosen large
enough such that all necessary derivatives of \(u\)  vanish at the domain
boundaries with numerical precision. However, the same approach for
initial data which do not tend to zero or are only slowly decreasing to zero for
\(x\rightarrow \infty \) , would in general imply a Gibbs phenomenon at the
domain boundaries. The resulting method would therefore be only of
first order in the number of Fourier modes, making it impossible to
reach the high resolution necessary to treat e.g. rapid oscillations
in the solution. Therefore, such situations are typically treated on a finite interval. This leads to the problem of how to impose boundary conditions, so that inside the computational domain the solution is the same as if the computation was done on the whole real line.
Such boundary conditions are called transparent.
Bérenger [1]} introduced perfectly
matched layers (PML) in electrodynamics to address this problem by
extending the computational domain to layers glued to the domain
boundaries. Inside the layers, the equation under consideration is deformed
to a dissipative one, which is chosen to efficiently dissipate the
solution. Whereas this works
well for linear equations, examples for the nonlinear Schrödinger
equation, see [2]}, [3]}, showed that in a nonlinear
setting there will be back
reflections from the layers to the computational domain. For integrable equations exact transparent boundary
conditions (TBC) can be given, e.g. for the case of modified KdV see [4]} based
on [5]}. The problem with both PML and TBC is that they
in general require initial data with compact support within the
computational domain, thus limiting the class of solutions that can be studied.
The goal of the present paper is to establish a spectral numerical
approach for generalised KdV
equations (REF ) for initial data that are analytic on the whole real line,
that are slowly decreasing towards infinity, or are bounded there. This numerical approach exhibits spectral convergence on the whole
real line with a technique similar to [6]}.
